380 XV NORMED ALGEBRAS AND SPECTRAL THEORY
therefore the isomorphism above extends to an isomorphism of Hilbert spaces
r : Hg> -> Lg(S, m), such that Ug.(x) = T"^lM(x)T. Let us now show that
or equivalently that, if F e ^c(S')? the continuous operator M(F) (" multipli-
cation by F ") on the Hilbert space Lc(S, m) has norm equal to ||F|| . It follows
immediately from (13.12.5) that ||M(F)|| ^ ||F||. On the other hand, there
exists Xi e S' such that \F(xi)\ = \\F\\ (3.17.10); hence, for all e > 0, there exists
a compact neighborhood V of Xi in S' such that \F(x)\ ^ ||F|| — e for all
X € V. Since by hypothesis the support ofm is the whole of S, and since in all
cases Xi ^es *n the closure of S in S', we have m(V n S) = j cpVn s dm =
(N2(<py n s))2 > 0; also it is clear that \F<pv n si ^ (I|F|| - e)cpv n s , so that
and consequently ||M(F)|| ^ ||F|| — e. Since e > 0 was arbitrary, this proves
our assertion. Hence the isomorphism A • 1H , + Ug>(x)t-+A + jc extends by
continuity to an isometry of ^, onto ^C(S'). Having regard to the fact that
every character of ^C(S') is of the form Fh-»F(/) for some / e S' (15.3.7), we
conclude that S^/ = S' and hence that Sg, = S. It remains to show that mg, = m,
or equivalently that JF(x) dmg>(x) = J Fft) dm(x) for all functions F e Jf C(S).
But by virtue of (i) we have
and since the functions £, as z runs through A, form a total set in ^c(S), this
establishes the equality of the bounded measures x$ • mg, and £$ • m for all
x, y in A. On the other hand, we have seen earlier that there exists a function
^ = S Xi*i (where the xt belong to A) which does not vanish on Supp(F), so
that F = G# with G e «?f C(S). Consequently we have from above
I
=
J
=1
This completes the proof of the Plancherel-Godement theorem.gligible; but